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Permutation Group


A permutation group is a finite group G whose elements are permutations of a given set and whose group operation is composition of permutations in G. Permutation groups have orders dividing n!.

Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.

Permutation groups are represented in the Wolfram Language as a set of permutation cycles with PermutationGroup. A set of permutations may be tested to see if it forms a permutation group using PermutationGroupQ[l] in the Wolfram Language package Combinatorica` .

Conjugacy classes of elements which are interchanged in a permutation group are called permutation cycles.

Examples of permutation groups include the symmetric group S_n (of order n!), the alternating group A_n (of order n!/2 for n>=2), the cyclic group C_n (of order n), and the dihedral group D_n (of order 2n).


See also

Alternating Group, Cayley's Group Theorem, Cycle Index, Cyclic Group, Dihedral Group, Group, Netto's Conjecture, Permutation, Permutation Cycle, Permutation Graph, Permutation Involution, Symmetric Group, Transposition

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References

Cameron, P. Permutation Groups. New York: Cambridge University Press, 1999.Furst, M.; Hopcroft, J.; and Luks, E. "Polynomial Time Algorithms for Permutation Groups." In Proc. Symp. Foundations Computer Sci. IEEE, pp. 36-41, 1980.Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984.Skiena, S. "Permutation Groups." §1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 17-26, 1990.Wielandt, H. Finite Permutation Groups. New York: Academic Press, 1964.

Referenced on Wolfram|Alpha

Permutation Group

Cite this as:

Weisstein, Eric W. "Permutation Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PermutationGroup.html

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